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Quantifying sustainability: Resilience, efficiency and the return of information theory
http://lietaer.com/images/Ecological_Complexity_Final.pdf

Robert E. Ulanowicz a,*, Sally J. Goerner b, Bernard Lietaer c, Rocio Gomez d
a University of Maryland Center for Environmental Science, Chesapeake Biological Laboratory, Solomons, MD 20688-0038, USA
b Integral Science Institute, 374 Wesley Ct, Chapel Hill, NC 27516, USA
c Center for Sustainable Resources, University of California, 101 Giannini Hall, Berkeley, CA 94720-3100, USA
d School of Computer Science, University of Birmingham, Birmingham B15 2TT, United Kingdom

Published on line 29 November 2008
Keywords:
Apophasis
Ascendency
Biodiversity
Fitness
Information theory
Reserve capacity
Resilience
Stability
Sustainability
Window of vitality

a b s t r a c t

Contemporary science is preoccupied with that which exists; it rarely accounts for what is
missing. But often the key to a system’s persistence lies with information concerning
lacunae. Information theory (IT), predicated as it is on the indeterminacies of existence,
constitutes a natural tool for quantifying the beneficial reserves that lacunae can afford a
system in its response to disturbance. In the format of IT, unutilized reserve capacity is
complementary to the effective performance of the system, and too little of either attribute
can render a system unsustainable. The fundamental calculus of IT provides a uniform way
to quantify both essential attributes – effective performance and reserve capacity – and
results in a single metric that gauges system sustainability (robustness) in terms of the
tradeoff allotment of each. Furthermore, the same mathematics allows one to identify the
domain of robust balance as delimited to a ‘‘window of vitality’’ that circumscribes
sustainable behavior in ecosystems. Sensitivity analysis on this robustness function with
respect to each individual component process quantifies the value of that link ‘‘at the
margin’’, i.e., how much each unit of that process contributes to moving the system towards
its most sustainable configuration. The analysis provides heretofore missing theoretical
justification for efforts to preserve biodiversity whenever systems have become too streamlined
and efficient. Similar considerations should apply as well to economic systems, where
fostering diversity among economic processes and currencies appears warranted in the face
of over-development.

1. Introduction: the importance of being absent

The late Bateson (1972) observed that science deals overwhelmingly
with things that are present, like matter and
energy. One has to dig deeply for exceptions in physics that
address the absence of something (like the Pauli Exclusion
Principle, or Heisenberg’s uncertainty). Yet any biologist can
readily point to examples of how the absence of something
can make a critical difference in the survival of a living system.
Nonetheless, because biology aspires to becoming more like
physics, very little in quantitative biology currently addresses
the important roles that lacunae play in the dynamics of living
systems.
One might object that the use of information theory (IT) in
genomics does indeed address matters like missing alleles, but
the emphasis in bioinformatics remains on information as a
positive essence—as something that is transferred between a
sender and a receiver. Biology at large has yet to reckon with
Bateson’s insight that IT addresses apophasis directly as
something even more fundamental than communication
theory. In particular, IT is a means for apprehending and
quantifying that which is missing. At the same time Bateson
made his penetrating observation, he also defined information
as any ‘‘difference that makes a difference’’, and such
difference almost always involves the absence of something.
All too many investigators, and even some theoreticians of
information, remain unaware that IT is predicated primarily
upon the notion of the negative. That this was true from the
very beginning can be seen in Boltzmann’s famous definition
of surprisal,


where s is one’s surprisal at seeing an event that occurs with
probability p, and k is an appropriate (positive) scalar constant.
Because the probability, p, is normalized to a fraction between
zero and one, most offhandedly conclude that the negative
sign is a mathematical convenience to make s work out
positive (and that may have been Boltzmann’s motivation).
But from the perspective of logic one can only read this
equation as defining s to gauge what p is not. That is, if p is
the weight we give to the presence of something, then s
becomes a measure of its absence.1 If p is very small, then
the ensuing large magnitude of s reflects the circumstance
that most of the time we do not see the event in question.
Boltzmann’s gift to science – the feasibility of quantifying
what is not – remains virtually unappreciated. It is akin to the
contribution of the Arabian mathematicians who invented the
number zero. (Anyone who doubts the value of that device
should try doing long division using Roman numerals.) In
particular, we shall attempt to build upon Boltzmann’s
invention and to demonstrate that IT literally opens new
vistas to which classical physics remains blind. More
importantly, the interplay between presence and absence
plays a crucial role in whether a system survives or disappears.
As we shall see, it is the very absence of order (in the form of a
diversity of processes) that makes it possible for a system to
persist (sustain itself) over the long run.
2. Evolution and indeterminacy
That Boltzmann’s definition is actually a quantification of the
negative lends an insight into IT that few appreciate—namely,
that the product of the measure of the presence of an event, i,
( pi) by a magnitude of its absence (si) yields a quantity that
represents the indeterminacy (hi) of that event,


When pi  1, the event is almost certain, and hi  0; then
when pi  0, the event is almost surely absent, so that again
hi  0. It is only for intermediate, less determinate values of
pi that hi remains appreciable, achieving its maximum at
pi = (1/e).
It is helpful to reinterpret (2) in terms germane to
evolutionary change and sustainability. When pi  1, the
event in question is a virtual constant in its context and
unlikely to change (hi  0). Conversely, whenever pi  0, the
event exhibits great potential to change matters (si 1), but it
hardly ever appears as a player in the system dynamics (so
that, again, hi  0). It is only when pi is intermediate that the
event is both present frequently enough and has sufficient
potential for change. In this way, hi represents the capacity for
event i to be a significant player in system change or evolution.
Seeking a perspective on the entire ensemble of events
motivates us to calculate the aggregate systems indeterminacy,
H, as
H
X
i
hi ¼ k
X
i
pi logð piÞ; (3)
which we can now regard as a metric of the total capacity of
the ensemble to undergo change. Whether such change will be
coordinated or wholly stochastic depends upon whether or
not the various events i are related to each other and by how
much. In order for any change to be meaningful and directional,
constraints must exist among the possible events
(Atlan, 1974).
In order better to treat relationships between events, it is
helpful to consider bilateral combinations of events, which for
clarity requires two indices. Accordingly, we will define pij as
the joint probability that events i and j co-occur. Boltzmann’s
measure of the non-occurrence of this particular combination
of events (1) thus becomes,
If events i and j are entirely independent of each other, the
joint probability, pij, that they co-occur becomes the product of
the marginal probabilities that i and j each occur independently
anywhere. Now, the marginal probability that i occurs
for any possible j is pi: ¼
P
j pi j, while the likelihood that j
occurs regardless of i is p: j ¼
P
i pi j.2 Hence, whenever i and j
are totally independent, pij = pi.p.j. Here the assumption is
made that the indeterminacy sij is maximal when i and j are
totally independent. We call that maximum s
i j. The difference
by which s
i j exceeds sij in any instance then becomes a
measure of the constraint that i exerts on j, call it xijj, where,
xij j ¼ s
i j  si j ¼k logð pi: p: jÞ  ½k logð pi jÞ ¼ k log
pi j
pi: p: j
!
¼ xjji:
(4)
The symmetry in (4) implies that the measure also describes
the constraint that j exerts upon i. In other words (4) captures
the mutual constraint that i and j exert upon each other (an
analog of Newton’s Third Law of motion).
In order to calculate the average mutual constraint (X)
extant in the whole system, one weights each xijj by the joint
1 Here the reader might ask why the lack of i is not represented
more directly by (1 – pi)? The advantage and necessity of using the
logarithm will become apparent presently.
2 For the remainder of this essay a dot in the place of an index
will represent summation over that index.
28 e c o l o g i c a l c omp l e x i t y 6 ( 2 0 0 9 ) 2 7 – 3 6
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probability that i and j co-occur and sums over all combinations
of i and j:
X ¼
X
i; j
pi jxij j ¼ k
X
i; j
pi j log
pi j
pi: p: j
!
(5)
Here is where the advantage of (1) as the formal estimate of
lacunae becomes apparent, because the convexity of the
logarithmic function guarantees (Abramson, 1963) that:
HX0 (6)
In words, (6) says that the aggregate indeterminacy is an
upper bound on how much constraint (order) can appear in a
system. Most of the time, H > X, so that the difference
C ¼ ðH  XÞ ¼ k
X
i; j
pi j log
p2
i j
pi: p: j
!
0 (7)
aswell. In the jargon of IT C is called the ‘‘conditional entropy’’.
Relationship (7) can be rewritten as
H ¼ X þ C; (8)
and it makes a very valuable statement. It says that the
capacity for evolution or self-organization (H) can be decomposed
into two components. The first (X) quantifies all that is
regular, orderly, coherent and efficient. It encompasses all
the concerns of conventional science. By contrast, C represents
the lack of those same attributes, or the irregular,
disorderly, incoherent and inefficient behaviors. It quantifies
and brings into the scientific narrative a host of behaviors
that heretofore had remained external to scientific discourse.
Furthermore, it does so in a way that is wholly
commensurate with X, the usual object of interest. The
key point is that, if one is to address the issues of persistence
and sustainability, C becomes the indispensable focus of
discussion, because it represents the reserve that allows
the system to persist (Conrad, 1983). To help one see this,
it is useful to demonstrate how onemight attach numbers to
these quantities.
3. Measuring the missing
Up to this point we have spoken only vaguely about events i
and j. Without loss of generality, we now narrow our
discussion to consider only transfers or transformations. That
is, event i will signify that some quantum of medium leaves or
disappears from component i. Correspondingly, event j will
signify that a quantum enters or appears in component j. We
now identify the aggregation of all quanta both leaving i and
entering j during a unit of time – or, alternatively, the flow from
i to j (or the transformation of i into j) – as Tij. Thus, Tij might
represent the flow of electrons from point i to point j in an
electrical circuit; the flow of biomass from prey i to predator j
in an ecosystem; or the transfer of money from sector i to
sector j in an economic community.
We maintain the convention introduced earlier that a dot in
the place of a subscript indicates summation over that index.
Thus Ti: ¼
P
jTi j
 
will represent everything leaving i during the
unit timeinterval, and T.jwill gauge everything entering jduring
the same duration. In particular, T:: ¼
P
i; jTi j
 
represents the
total activity of the system and is given the name ‘‘total system
throughput’’.
These definitions allow us to estimate all the probabilities
defined above in terms of their measured frequencies of
occurrence. That is,
pi j 
Ti j
T::
; pi: 
Ti:
T::
; and p: j 
T: j
T::
(9)
Substituting these estimators in Eqs. (3), (5) and (7), yields
H ¼ k
X
i; j
Ti j
T::
log
Ti j
T::
 
; X ¼ k
X
i; j
Ti j
T::
log
Ti jT::
Ti:T: j
!
; and
C ¼ k
X
i; j
Ti j
T::
log
T2
i j
Ti:T: j
!
; (10)
respectively.
The dimensions in the definitions (10) remain problematic,
however. All of the ratios that occur there are dimensionless (as
required of probabilities), so that the only dimensions that the
variables H, X and C carry are those of the base of the logarithm
used in their calculation. For example, if the base of the
logarithm is 2, the variables are all measured in bits. (A ‘‘bit’’ is
the amount of information required to resolve one binary
decision.) Unfortunately, bits do not convey any sense of the
physical magnitude of the systems to which they pertain. For
example,anetworkofflowsamong thepopulationsofmicrobes
in a Petri Dish could conceivably yield an H of the same order of
magnitude as a network of trophic exchanges among the
mammalian species on the Serengeti Plain.
Tribus and McIrvine? (1971) spoke to this inadequacy of
information indices and suggested that the scalar constant, k,
which appears in each definition, be used to impart physical
dimensions to themeasures.Accordingly,we elect to scale each
index by the total system throughput, T.., which conveys the
overall activity of the system. In order to emphasize the new
nature of the results, we give them all new identities. We call
C ¼ T:: H ¼ 
X
i; j
Ti j log
Ti j
T::
 
(11)
the ‘‘capacity’’ for system development (Ulanowicz and Norden,
1990). The scaled mutual constraint,
A ¼ T:: X ¼
X
i; j
Ti j log
Ti jT::
Ti:T: j
!
; (12)
we call the system ‘‘ascendency’’ (Ulanowicz, 1980). The
scaled conditional entropy,
F ¼ T:: C ¼ 
X
i; j
Ti j log
T2
i j
Ti:T: j
!
; (13)
we rename the system ‘‘reserve’’,3 for reasons that soon
should become apparent.
3 From here on ‘‘reserve’’ will apply to what heretofore has been
called ‘‘reserve capacity’’.


4. A two-tendency world
Of course, this uniform scaling does not affect the decomposition
(8), which now appears as
C ¼ A þ F: (14)
In other words, (14) says that the capacity for a system to
undergo evolutionary change or self-organization consists of
two aspects: It must be capable of exercising sufficient
directed power (ascendency) to maintain its integrity over
time. Simultaneously, it must possess a reserve of flexible
actions that can be used to meet the exigencies of novel
disturbances. According to (14) these two aspects are literally
complementary. That they are conceptually complementary
as well is suggested by the following example:
Fig. 1 depicts three pathways of carbon flow (mg Cm2 y1)
in the ecosystem of the cypress wetlands of S. Florida that lead
from freshwater shrimp (prawns) to the American alligator via
the intermediate predator categories—turtles, large fish and
snakes (Ulanowicz et al., 1996).
Of course, these species are entwined in a myriad of
relationships with other populations, but for the purposes of
illustrating a point, this sub-network will be considered as if it
existed in isolation. T.. for this system is 102.6 mg C m2 y1;
the ascendency, A, works out to 53.9 mg C-bitsm2 y1 and
the reserve, F, is 121.3 mg C-bitsm2 y1.
Inspection of the pathways reveals that the most efficient
pathway between prawns and alligators is via the large fishes.
If efficiency were the sole criterion for development, the route
via the fish would grow at the expense of the less efficient
pathways until it dominated the transfer, as in Fig. 2.
The total system throughput of the simplified system rises
to 121.8 mg C m2 y1, as a result of the increase in overall
efficiency, but the greatest jump is seen in the ascendency, A,
which almost doubles to 100.3 mg C-bitsm2 y1. Meanwhile,
the reserve has vanished completely (F = 0). To use a cliche´ ,
the system has put all its eggs in one basket (efficiency).
Should some catastrophe, like a virus affecting fish, devastate
the fish population, all transfer from prawns to alligators in
this rudimentary example would suffer in direct proportion.
If healthy populations of turtles and snakes had been
present when the fish population was incapacitated, it is
possible that the pathways they provide might have buffered
the loss, as in Fig. 3.
Rather than total system collapse, T.. drops modestly to
99.7 mg Cm2 y1, and the ascendency falls back only to
44.5 mg C-bitsm2 y1. The chief casualty of the disappearance
of the fishes is the reserve, which falls by almost half to
68.2 mg C-bitsm2 y1. In other words, in the alternative
scenario the system adapts in homeostatic fashion to buffer
performance (A) by expending reserves (F) (Odum, 1953). The
reserve in this case is not some palpable storage, like a cache of
some material resource. Rather, it is a characteristic of the
system structure that reflects the absence of effective
performance.
The hypothetical changes in Figs. 1–3 were deliberately
chosen as extremes to make a didactic point. In reality, one
might expect some intermediate accommodation between A
andFas the outcome. Identifying where such accommodation
might lie is the crux of this essay, for it becomes obvious that
the patterns we see in living systems are the outcomes of two
antagonistic tendencies (Ulanowicz, 1986, 1997). On one hand
are those processes that contribute to the increase in order
and constraint in living systems. Paramount among them
seems to be autocatalysis, which is capable of exerting
selection pressure upon its constituents and of exerting a
centripetal pull upon materials and energy, drawing resources
into its orbit (Ulanowicz, 1986, 1997). In exactly the opposite
direction is the slope into dissipation that is demanded by the
second law of thermodynamics. At the focal level, these trends
are antagonistic. At higher levels, however, the attributes
become mutually obligate: A requisite for the increase in
effective orderly performance (ascendency) is the existence of
flexibility (reserve) within the system. Conversely, systems
that are highly constrained and at peak performance (in the
second law sense of the word) dissipate external gradients at
ever higher gross rates (Schneider and Kay, 1994; Ulanowicz,
2009).

Fig. 1 – Three pathways of carbon transfer (mg CmS2? yS1)
between prawns and alligators in the cypress wetland
ecosystem of S. Florida (Ulanowicz et al., 1996).
Fig. 2 – The most efficient pathway in Fig. 1 after it had
eliminated parallel competing pathways.
Fig. 3 – Possible accommodation by turtles and snakes to
the disappearance of fish as intermediaries between
prawns and alligators.


5. The survival of the most robust
While the dynamics of this dialectic interaction can be quite
subtle and highly complex, one thing is boldly clear—systems
with either vanishingly small ascendency or insignificant
reserves are destined to perish before long. A system lacking
ascendency has neither the extent of activity nor the internal
organization needed to survive. By contrast, systems that are
so tightly constrained and honed to a particular environment
appear ‘‘brittle’’ in the sense of Holling (1986) or ‘‘senescent’’ in
the sense of Salthe (1993) and are prone to collapse in the face
of even minor novel disturbances. Systems that endure – that
is, are sustainable – lie somewhere between these extremes.
But, where?
Recognizing the importance of achieving an intermediate
balance, Wilhelm (2003) suggested that the product AF might
serve as an appropriate metric for robustness. It becomes zero
whenever either A or F is zero and it takes on a maximum
when A = F. This is an interesting suggestion, and strongly
parallels the treatment of power production by Odum and
Pinkerton (1955); but, like the latter analysis, it remains
problematic. For example, there is no obvious reason why the
optimal balance should fall precisely at A = F (other than
mathematical convenience). Secondly, with regard to this
analysis, the simple product AF does not accord with the
formulaic nature of the IT used to construct each of its factors.
This second shortcoming is rather easy to rectify once we
recall that (2) behaves in much the same manner as AF. It is
zero at the extremes of pi = 0 and pi = 1 and achieves a single
maximum in between. Accordingly, we define a = A/C and
notice that 1 > a > 0. Here a is a relative measure of the
organized power flowing within the system. In lieu of a, or
(1  a), we choose the Boltzmann formulation, –k log(a), so that
the product of a and a, or what we shall call the system’s
‘‘fitness for evolution’’,
F ¼ ka logðaÞ; (15)
becomes our measure of the system’s potential to evolve or
self-organize. It is 0 for a = 1 and approaches the limit of 0 as
a !0. One can normalize this function by choosing k = e log(e)
(where ‘‘e’’ is the base of natural logarithms), such that
1 > F > 0.
This does not solve our second problem, however, as F is
still constrained to peak at a = (1/e). There is no more reason to
force the balance between A and F to occur at [A/(A + F)] = (1/e)
than it was to mandate that it happen when A = F. Clearly, the
location of the optimum could be the consequence of (as yet)
unknown dynamical factors, rather than one of mathematical
convenience. One way to permit the maximum to occur at an
arbitrary value of a is to introduce an adjustable parameter,
call it b, and to allow the available data to indicate the most
likely value of b. Accordingly, we set F = –kab log(ab). This
function can be normalized by choosing k = e/log(e), so that
Fmax = 1 at a = e1/b, where b can be any positive real number.
Whence, our measure for evolutionary fitness becomes
F ¼ 
e
logðeÞ
 
ab logðabÞ (16)
The function F varies between 0 and 1 and is entirely
without dimensions. It describes the fraction of activity that is
effective in creating a sustainable balance between A and F.
That is, the total activity (e.g., the GDP in economics, or T.. here)
will no longer be an accurate assessment of the robustness of
the system. Our measure, T.., must be discounted by the
fraction (1  F). Equivalently, the robustness, R, of the system
becomes
R ¼ T:: F: (17)
The focus of attention now turns to identifying the most
propitious value for b. This is a very crucial point, because the
value of b fixes the optimal value of a against which the status
of any existing network will be reckoned. There is no apriori
reason to assume that the value of b is universal. There might
be one value of b most germane to ecosystem networks,
another for economic communities, and still another for
networks of genetic switching. Since the data most familiar to
the authors of this work pertain to ecosystem networks of
trophic exchanges, ecology seems a reasonable domain in
which to begin our search.
Data on existing flow networks of ecosystems do not
appear sufficient to determine a precise value for b. They do,
however, indicate rather clearly those configurations of flows
that are not sustainable. Zorach and Ulanowicz (2003), for
example, compare how a collection of estimated flow
structures differs from networks that have been created at
random. For their demonstration, they plotted the networks,
not on the axes A vs. F, but rather on the transformed axes
c = 2F/2 and n = 2A. As they explain in the course of their
analysis, c measures the effective connectivity of the system in
links per node, or how many nodes on (logarithmic) average
enter or leave each compartment. The variable n gauges the
effective number of trophic levels in the system, or how many
transfers, on (logarithmic) average, a typical quantum of
medium makes before leaving the system. Their results are
displayed in Fig. 4.
It is immediately obvious that the empirical networks all
cluster within a rectangle that is bounded roughly in the
vertical direction by c = 1 and c  3.01 and horizontally by n = 2
and n  4.5. It happens that three of the four sides of this
‘‘window of vitality’’ can be explained heuristically. The fact
that c  1 says simply that the networks being considered are
all fully connected. Any value c < 1 would imply that the graph
Fig. 4 – Combinations of link-density (c) plotted against
number of effective roles (n) in a set of randomly
assembled networks (circles) and empirically estimated
ecosystem networks (dark squares).
e c ologi c a l c o m p l e x i t y 6 ( 20 0 9 ) 2 7 – 36 31
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is separated into non-communicating sub-networks. Similarly,
n > 2 for all ecosystemnetworks, because it is in the very
definition of an ecosystem that it encompass complementary
processes, such as oxidation/reduction reactions or autotrophy/
heterotrophy interactions (Fiscus, 2001).
The boundary delimiting maximal link-density, c  3.01, is
the result of applying the May–Wigner stability hypothesis in
its information-theoretic homolog (Ulanowicz, 2002). The
precise theoretical value of the boundary, as derived by
Ulanowicz, is c = e(3/e). In essence, this says that systems can be
either strongly connected across a few links or weakly
connected across many links, but configurations of strong
connections across many links and weak connections
across a few links tend to break up or fall apart, respectively
(May, 1972). The ‘‘magic number’’ 3 in association with
maximal connectivity has been cited by Pimm (1982) and by
Wagensberg et al. (1990) for ecosystems, and by Kauffman
(1991) for genetic networks.
Only the fourth boundary remains largely unexplained.
Pimm and Lawton (1977) commented on how one rarely
encounters trophic pathways greater than 5 in nature. Efforts
to relate this limit to thermodynamic efficiencies have
(thus far) proved unsuccessful (Pimm and Lawton, 1977).
The available data reveal no values for n close to 5, and so an
upper limit of 4.5 has been chosen arbitrarily.
The emerging picture seems to be that sustainable
ecosystems all plot within the window of vitality. It has yet
to be investigated whether any sub-regions of the window
might be preferred over others, and the scatter appears to be
without statistically discernible pattern. It might be surmised,
however, that systems plotting too close to any of the four
boundaries could be approaching their limits of stability for
one reason or another. Under such consideration, the most
conservative assumption would be that those systems most
distant from the boundaries are those most likely to remain
sustainable. We therefore choose the geometric center of the
window (c = 1.25 and n = 3.25) as the best possible configuration
for sustainability under the information currently available.
These values translate into a = 0.4596, from which we
calculate a most propitious value of b = 1.288.
6. Vectors to sustainability
Systems can risk unsustainability in relation to this ‘‘optimum’’
on two accounts. When a < 0.4596, the system likely requires
more coherence and cohesion. There may be insufficient or
under-developed autocatalytic pathways that could impart
additional robustness to the system. Conversely, when
a > 0.4596, the system might be over-developed or too tightly
constrained. Some autocatalytic pathwaysmay have arrogated
too many resources into their orbit, leaving the system with
insufficient reserves to persist in the face of novel exigencies.
Should it survive further scrutiny, this threshold in a
provides an extremely useful guide towards achieving
sustainable communities. In fact, the measure of robustness,
R, can even be employed to indicate which features of a given
configuration deserve most remediation. Once again, the
algebra of IT proves most convenient, because the functions C,
A and F all happen to be homogeneous Euler functions of the
first order. This means that the derivatives with respect to
their independent variables are relatively easy to calculate
(Courant, 1936).
Starting from our definition of robustness (17), we seek to
establish the direction in which this attribute responds to a
unit change in any constituent flow. That is we wish to
calculate (@R/@Tij). Employing the chain rule of differentiation,
we see that
@R
@Ti j
¼ F þ T::
@F
@Ti j
@R
@Ti j
¼ F þ T::F0 @a
@Ti j
@R
@Ti j
¼ F þ
T::F0
C
log
Ti jT::
Ti:T: j
" #
þ a log
T2
i j
Ti:T: j
( " #)
(18)
where F0 is the derivative of F with respect to a, i.e.,
F0 ¼ ebab1 logðabÞ
logðeÞ
þ 1
 
(19)
In particular, when the system is at its optimum
(F = 1 and F0 = 0) we see from (18) that a unit increment in
each and every flow in the system would contribute exactly
one unit to system robustness. Once away from the optimum,
however, contributions at the margin will depend on which
side of the optimum the system lies, and where in the network
any particular contribution is situated.
When a < aopt, then F0 will be positive, so that those flows
that dominate the inputs to or output from any compartment
will result in a positive sum within the braces, and the
contribution of that transfer at the margin will be >1. For the
relatively smaller flows, the negative second term in braces
will dominate, and the contribution of those links at the
margin will be <1. One observes both situations within the
network of energy flows occurring in the Cone Spring
ecosystem (Tilly, 1968), one of the most widely used examples
of a simple ecosystem flow network (Fig. 5). The value of a for
this network (0.418) is <aopt (0.460), so that the community
can still grow and develop without jeopardizing its sustainability.
Those flows with highest contributions at the margin
(in parentheses) serve to vector the system towards configurations
of greater sustainability. One notes in particular that
increases in the values of the contributions at the margin
along the pathway ! 1! 2! 3 are all favored to move a
towards aopt. Conversely, increases in flows that parallel
mainstream flows (such as 3! 4) contribute proportionately
much less than towards system robustness, so that there is a
disincentive against augmenting those flows.
To demonstrate that increases along the pathway
! 1! 2! 3! will indeed raise the value of the relative
ascendency, we artificially add, say 8000 kcal m2 y1, to each
of those links (Fig. 6). These additions mimic the process of
eutrophication, whereby the addition of some additional
resource inflates the primary production of the plants [1],
which primarily die uneaten to become detritus [2], which in
its turn is consumed and dissipated by bacteria [3]. The
ensuing value of a is 0.529 (>aopt).

The absolute values of the contributions at the margin in
Fig. 6 are pretty much the qualitative inverses of those in Fig. 5.
One sees, for example, that the contributions at the margin
along the ‘‘eutrophic axis’’ are now all less than one, whereas
the corresponding values of the small, parallel transfers (such
as 3 !4) are now significantly greater than unity. One
concludes, not surprisingly, that in a system with a surfeit
of ascendency over reserve, system survival is abetted by the
addition of small, diverse parallel flows.
7. One-eyed ecology
One may conclude several things from the model developed
here, but one in particular stands out: Many ecologists, in their
desire for a science that is derivative of physics, have
unnecessarily blinded themselves to much of what transpires
in nature. Physics does address matter and energy as they are
present in ecosystems, but it tells us almost nothing about that
which is lacking. Such latter considerations remain external to
the core dynamics and can only be accounted as boundary
constraints in ad-hoc terms, such as ‘‘rules’’ (Pattee, 1978) or
particular ‘‘material laws’’ (Salthe, 1993), that usually do not
conveniently mesh with the formal structure or dimensionality
of the primary description.
As we have seen, the notions of both presence and absence
are built into the formal structure of IT. Such architecture
accounts for relationships like (8) and (14) wherein complementary
terms of ‘what is’ and ‘what is not,’ share the same
dimensions and almost the same structure. That is, one is
Fig. 5 – The trophic exchanges of energy (kcalmS2 yS1) in the Cone Spring ecosystem (Tilly, 1968). Arrows not originating
from a box represent exogenous inputs. Arrows not terminating in a box portray exogenous outputs. Ground symbols
represent dissipations. Numbers in parentheses accompanying each flow magnitude indicate the value of 1 kcalmS2 yS1
increment at the margin.
Fig. 6 – The Cone Spring network, as in Fig. 5, except that 8000 kcalmS2 yS1 has artificially been added over the route,
!1 !2! 3!. Corresponding changes in the increments at the margin are shown in parentheses.

comparing apples with apples. Furthermore, the effects of
lacunae no longer remain external to the statement of the
dynamics; they become central to it. Most importantly, by
incorporating apophasis into the core of the problem, one can
avoid the pursuit of ill-fated directions, as will be discussed
presently.
In fairness, it should be recalled that ecologists were not
always indifferent to IT. In fact, soon after Claude Shannon
(1948) had resuscitated Boltzmann’s (1872) formulation,
Robert MacArthur? (1955) used the index to quantify the
diversity of flows in an ecosystem and suggested that such
diversity enhanced the stability of an ecosystem. Unfortunately,
focus soon switched away from flows to the diversity of
populations, and the leading aspiration among theoretical
ecologists during the decade of the 1960s became how to
demonstrate that biodiversity augments system stability
(which in the interim had been formulated in terms of linear
dynamical theory (Ulanowicz, 2002)). To the chagrin of most
who were pursuing this intuition, physicist May (1972) upset
the applecart by showing how, under the assumption of
random connections, increased diversity is more likely to
decrement, rather than bolster, system stability. May’s
counter-demonstration proved an embarrassment of the first
magnitude to ecologists, while at the same time reinforcing
their physics envy (Cohen, 1976). As a result, most ecologists
came to eschew IT, and retreated to approaches that
resembled what remained orthodox in physics.
We now discern a larger vision of the diversity/stability
issue. It is not that May was wrong in his elegant demonstration.
Rather (referring to Fig. 4), May showed how, as a system
approached the top frame of the window of vitality, further
diversification will indeed accelerate the system across the
threshold and into oblivion. Furthermore, May’s stability
index was pivotal to establishing the location of this upper
frame. The system is over-connected (see Allen and Starr,
1982), but we now see that transgressing May’s threshold is
only one of four different ways that a system can get into
trouble. A system could also approach the bottom frame
(under-connected), at which time, according to the model just
presented, further diversification indeed will reduce the
tendency of the system to become unsustainable. This latter
argument, however, requires an appropriate measure of the
Reserve that will keep the system from approaching that edge
too closely. Of course, a system could also exit the window via
the end members, but the exact location of the right-hand
limit and the reasons for its existence remain poorly understood.
Most fortunately,May’s demonstration did little to quench
the widely held conviction that biodiversity does have value
in maintaining sustainable ecosystems. Major worldwide
efforts have been justifiably mounted to conserve ecological
diversity.Yet, although some empirical evidence does exist to
support such intuition (e.g., Van Voris et al., 1980; Tilman
et al., 1996), few convincing theoretical models have emerged
to defend such conservation. Few, that is, save for that of
Rutledge et al. (1976), who focused undauntedly on the utility
of IT and in particular upon the conditional entropy as an
index of merit. Unfortunately, the ecologists of Rutledge’s
time were in no mood to countenance a return to the
shambles of IT that lay in the wake of May’s deconstruction.
Instead, ecology marched on with one eye kept deliberately
shut.

8. Evolution as moderation

The model just discussed highlights the necessary role of
reserve capacities in sustaining ecosystems. It contrasts with
Darwinian theory, which unfortunately is espoused by many
simply as the maximization of efficiency4 (e.g., the survival of
the fittest). Such emphasis on efficiency is evident as well in a
number of approaches to ecology, such as optimal foraging
theory.5 Our results alert us to the need to exhibit caution
when it comes to maximizing efficiencies. Systems can
become too efficient for their own good. Autocatalytic
configurations can expand to suck away resources from
nonparticipating taxa, leaving them to wither and possibly
to disappear. In particular, the human population and its
attendant agro-ecology is fast displacing reserves of wild biota
and possibly driving the global ecosystem beyond aopt. In the
face of such monist claims, our model illustrates the pressing
need to conserve the diversity of biological processes (which,
after all, was MacArthur?’s original concern).
Although possibly less enamored of physics, economics,
too, seems in pursuit of monistic goals and all too willing to
sacrifice everything for the betterment of market efficiency.
Doubtless, maximizing efficiency is a good strategy to apply to
inchoate economic systems that occupy the upper-left-hand
corner of the economic window of vitality. Preoccupation
with efficiency in today’s global theatre could, however,
propel into disaster a global economy that is fast approaching
the lower-right-hand corner. Economists have long recognized,
usually for ethical reasons, the need for ‘‘externalities’’
to brake the pell-mell rush towards increased market
‘‘efficiencies.’’ In the model presented here, such brakes
appear as necessary internal constraints on the system and
point up the need to retain ‘‘subsidiarity.’’ At times the brakes
appear spontaneously within the system, as when societies
adjust to problems by increasing their complexity (Tainter,
1988). Our model suggests that the establishment of complementary
currencies canmake impressive contributions at
the margin towards sustaining the global economic system
(Lietaer, 2001).
While this exercise may have illumined the dialectical
nature of natural development in consistent and quantitative
terms, it unfortunately leaves other issues clouded. The exact
location of aopt, for example, is bound to remain controversial.
Two major uncertainties further obscure this problem. The
first is the pressing need for a larger collection of ecological
flow networks with which to explore whether any sub-region
within the window of vitality may exist that is favored by the
most sustainable systems. In this context it is worth
remarking that almost all ecosystem networks plotted in
Fig. 4 for which a > aopt constitute early renditions of
4 Efficiency is being used here as a synonym for ‘‘effectiveness’’.
5 Optimal foraging theory has been criticized elsewhere for
focusing on the wrong null hypothesis. Allen et al. (2003) argue
that optimal foraging is not the signal of interest, but is rather the
null hypothesis, given evolved systems.

ecosystem flows that were cast in terms of only a few
compartments. Recent and more fully resolved networks of
flows tend to possess lower values of a (Robert Christian
personal communication). Christian further notes that the
values of a for the stable subset of a large collection of
weighted, randomly assembled networks approached an
asymptote very close to 1/e. (See Fig. 7.11a on p. 84 of Morris
et al., 2005.) It therefore bears further investigation as to
whether b differs significantly from unity. If b  1, then the aopt
chosen here is decidedly an over-estimate. This has practical
implications, because an inflated aopt would not sound
warning bells soon enough.
In addition, there are the nagging ambiguities concerning
the origins of the truncation of the right side of the window.
Our choice of n = 4.5 as the limit was, to a degree, arbitrary.
Unfortunately, little is yet known as to what poses a limit on
the effective number of trophic levels in ecosystems.
Obviously, thermodynamic losses play a key role, but they
do not seem to be the whole story (Pimm and Lawton, 1977).
Needed is a theoretical explanation of that limit akin to May’s
exegesis, which fixes the position of the top member. (Apropos
this limit is the observation by Cousins (1990) that it is the top
members of the trophic web that often control what transpires
at lower levels.)
Of course, there remains the question of how well (if at all)
this ecological analysis pertains to economic communities. It
seems not unreasonable to assume that many of the same
dynamics are at work in economics as structure ecosystems,
and that, over ‘‘deep time,’’ nature has solved many of the
developmental problems for ecosystems that still beset
human economies. It is perhaps useful that this model
suggests that Adam Smith’s ‘‘invisible hand’’ is not alone in
sculpting the patterns of economies. Yet another hand would
appear necessary to work opposite to Smith’s. (No one claps
with one hand.) Furthermore, it now appears unfortunate that
economists by and large have abandoned the study of ‘‘inputoutput’’
networks as puerile. Perhaps this exercise will
motivate a few to dust off some of the archived, large data
sets on input–output networks of cash flows to scope out
better the dimensions of the economic counterpart to
ecology’s window of vitality.
In all likelihood, the dynamics portrayed here pertain
to other domains of inquiry as well. Kauffman (1991),
for example, has written about limits on the stability of
genetic control networks. These systems appear more
mechanical than do ecosystems, and their rigidities may
narrow the window of vitality sufficiently that it could be
characterized more as an ‘‘edge’’. As with economics, a large
collection of genetic control networks might help resolve
better the domain of ontogenetic stability. Similarly,
Vaz and Carvalho (1994) have portrayed the immune system
as a network. Might not elucidating its window of vitality
provide significant new insights into the health of organisms?
The possibilities of this wider perspective are truly exciting.
Their potential helps to vanquish the pessimism implicit in
ecologists’ physics envy and to give new life to Hutchinson’s
optimistic view of ‘‘ecology as the study of the Universe’’ (Jolly,
2006). All that is necessary is that ecologists keep both eyes
wide open.
Acknowledgements
The authors are grateful to the Citerra Foundation of Boulder,
Colorado for partial support of the travel involved in creating
this work and to Edgar and Christine Cahn for hosting our
initial meeting in their home. They would also like to extend
their gratitude to Timothy F.H. Allen for his enthusiastic
support of the ideas contained herein. Stanley N. Salthe and
Robert R. Christian also contributed several points to the
narrative following their reviews of an early draft.
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